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<h1>Material Types</h1>

	
<p>NairnFEA and NairnMPM support a variety of material types. Details on each material are given on the OSUPDocs wiki for <a href="http://osupdocs.forestry.oregonstate.edu/index.php/Material_Models">MPM Materials</a>, for <a href="http://osupdocs.forestry.oregonstate.edu/index.php/FEA_Material_Models">FEA Materials</a>, and for <a href="http://osupdocs.forestry.oregonstate.edu/index.php/Traction_Laws">traction law materials</a>; this section just lists the available materials and the properties needed to use them in calculations.</p>

<p>A material or traction law is defined in scripting commands using a block that allows you to specify a new material type and define all its properties. The command block format is:</p>
	
<blockquote>Material (id),(name),(type)<br>
&nbsp;&nbsp;(Property commands - one per line)<br>
Done</blockquote>
	
<p>where <code>(id)</code> is the ID or Label for the material created by the command,  <code>(name)</code> is the material name entered as a string, and (type) is the material type, which is enter by material name or number in the <a href="#tables">table below.</a></p>

<p>The Material command should be followed by lines that contain the material properties. Each property should be on a separate line and when the last property is specified, enter a line with the single <code>Done</code> Command. To set a property, use the property name (case sensitive) followed by a space and then its value. For example, to set modulus of an isotropic material, use</p>

<blockquote>E 2300</blockquote>

<p>A few properties differ from the above (name,value) pair; those exceptions are in the material definitions.</p>

<h2><a name="tables"></a>Tables of Materials and other Lawss</h2>

<p>The following tables list <a href="#solidmats">material models</a>, <a href="#tractionlaws">traction laws</a>, <a href="#contactlaws">contact laws</a>, <a href="#initiation">damage initation laws</a>, and <a href="#softening">softening laws</a> that call can be created with the <code>Material</code> command. The first column is the <code>(type)</code> code for the material while the second is numerical eequivalent to that code. The columns on the right list the type of analyses that allow that material type (with an "X"). The 2D MPM column has three subcolumns or use of that material in plane stress (P&sigma;), plane strain (P&epsilon;), or axisymmetric (AS) calculations.  Clicking any material name will show a list of all properties that can be set by replacing "prop" in an subordinate element with the property name (case sensitive).</p>

<h3><a name="solidmats"></a>Linear Elastic Small Strain Materials</h3>

<table border="1" cellpadding="2" cellspacing="0" width="90%" align="center">
<tr><th rowspan="2">Name</th><th rowspan="2">Type</th><th rowspan="2">Description</th><th rowspan="2">FEA</th><th colspan="3">2D MPM</th><th rowspan="2">3D MPM</th>
</tr>
<tr><th>P&sigma;</th><th>P&epsilon;</th><th>AS</th>
</tr>

<tr>
<td><a href="#iso"><code>Isotropic</code></a></td>
<td>1</td>
<td>Linear elastic, isotropic</td>
<td align="center">X</td><td align="center">X</td><td align="center">X</td><td align="center">X</td><td align="center">X</td>
</tr>

<tr>
<td><a href="#trans1"><code>Transverse 1</code></a></td>
<td>2</td>
<td>Linear elastic, transversely isotropic with unique axis perpendicular to plane of the 2D analysis.</td>
<td align="center">X</td><td align="center">X</td><td align="center">X</td><td align="center">X</td><td align="center">X</td>
</tr>

<tr>
<td><a href="#trans2"><code>Transverse 2</code></a></td>
<td>3</td>
<td width="50%">Linear elastic, transversely isotropic with unique axis in plane of 2D analysis. Unless rotated by &quot;angle&quot; options, the unique material axis is parallel to the <code>y</code> axis for 2D analyses (or <code>Z</code> axis for axisymmetric analyses).</td>
<td align="center">X</td><td align="center">X</td><td align="center">X</td><td align="center">X</td><td align="center">X</td>
</tr>

<tr>
<td><a href="#ortho"><code>Orthotropic</code></a></td>
<td>4</td>
<td width="50%">Linear elastic, orthotropic material</td>
<td align="center">X</td><td align="center">X</td><td align="center">X</td><td align="center">X</td><td align="center">X</td>
</tr>

<tr>
<td><a href="#bistab"><code>Bistable</code></a></td>
<td>10</td>
<td width="50%">Elastic, isotropic material with two stable states and reversible or irreversible transitions between the two states</td>
<td align="center"></td><td align="center">X</td><td align="center">X</td><td align="center">X</td><td align="center"></td>
</tr>
</table>

<h3>Hyperelastic Materials</h3>

<table border="1" cellpadding="2" cellspacing="0" width="90%" align="center">
<tr><th rowspan="2">Name</th><th rowspan="2">Type</th><th rowspan="2">Description</th><th rowspan="2">FEA</th><th colspan="3">2D MPM</th><th rowspan="2">3D MPM</th>
</tr>
<tr><th>P&sigma;</th><th>P&epsilon;</th><th>AS</th>
</tr>

<tr>
<td><a href="#mooney"><code>Mooney</code></a></td>
<td>8</td>
<td width="50%">Mooney-Rivlin, hyperelastic material. This material includes neo-Hookean
and rubber elastic materials as special cases.</td>
<td align="center"></td><td align="center">X</td><td align="center">X</td><td align="center">X</td><td align="center">X</td>
</tr>

<tr>
<td><a href="#neoh"><code>Neo-Hookean</code></a></td>
<td>28</td>
<td width="50%">Neo-Hookean, hyperelastic material.</td>
<td align="center"></td><td align="center">X</td><td align="center">X</td><td align="center">X</td><td align="center">X</td>
</tr>

<tr>
<td><a href="#idealgas"><code>IdealGas</code></a></td>
<td>22</td>
<td width="50%">Ideal Gas, as a hyperelastic material</td>
<td align="center"></td><td align="center"></td><td align="center">X</td><td align="center">X</td><td align="center">X</td>
</tr>

<tr>
<td><a href="#heaniso"><code>HEAnisotropic</code></a></td>
<td>21</td>
<td width="50%">Anisotropic, elastic-plastic, hyperelastic material (in development).</td>
<td align="center"></td><td align="center"></td><td align="center"></td><td align="center"></td><td align="center">X</td>
</tr>

<tr>
<td><a href="#tait"><code>TaitLiquid</code></a></td>
<td>22</td>
<td width="50%">Newtonian liquid with Tait equation for pressure as hyperelastic material.</td>
<td align="center"></td><td align="center"></td><td align="center">X</td><td align="center">X</td><td align="center">X</td>
</tr>

</table>

<h3>Elastic-Plastic Small Strain Materials</h3>

<table border="1" cellpadding="2" cellspacing="0" width="90%" align="center">
<tr><th rowspan="2">Name</th><th rowspan="2">Type</th><th rowspan="2">Description</th><th rowspan="2">FEA</th><th colspan="3">2D MPM</th><th rowspan="2">3D MPM</th>
</tr>
<tr><th>P&sigma;</th><th>P&epsilon;</th><th>AS</th>
</tr>

<tr>
<td><a href="#isoplast"><code>IsoPlasticity</code></a></td>
<td>9</td>
<td width="50%">Small-strain, elastic-plastic material with a hardening law</td>
<td align="center"></td><td align="center">X</td><td align="center">X</td><td align="center">X</td><td align="center">X</td>
</tr>

<tr>
<td><a href="#hill"><code>HillPlastic</code></a></td>
<td>15</td>
<td width="50%">Elastic-plastic, orthotropic material with a Hill yield criterion and hardening</td>
<td align="center"></td><td align="center"></td><td align="center">X</td><td align="center">X</td><td align="center">X</td>
</tr>

</table>

<h3><a name="SSSM"></a>Softening Small Strain Materials</h3>

<table border="1" cellpadding="2" cellspacing="0" width="90%" align="center">
<tr><th rowspan="2">Name</th><th rowspan="2">Type</th><th rowspan="2">Description</th><th rowspan="2">FEA</th><th colspan="3">2D MPM</th><th rowspan="2">3D MPM</th>
</tr>
<tr><th>P&sigma;</th><th>P&epsilon;</th><th>AS</th>
</tr>

<tr>
<td><a href="#isosoft"><code>IsoSoftening</code></a></td>
<td>50</td>
<td width="50%">Small-strain isotropic material with anisotropic damage mechanics using mode I and mode I softening laws.</td>
<td align="center"></td><td align="center">X</td><td align="center">X</td><td align="center">X</td><td align="center">X</td>
</tr>

<tr>
<td><a href="#transisosoft"><code>TransIsoSoftening 1</code></a></td>
<td>51</td>
<td width="50%">Small-strain transversely material with anisotropic damage mechanics using four softening laws. Unrotated axial direction along the z axis.</td>
<td align="center"></td><td align="center">X</td><td align="center">X</td><td align="center">X</td><td align="center">X</td>
</tr>

<tr>
<td><a href="#transisosoft"><code>TransIsoSoftening 2</code></a></td>
<td>51</td>
<td width="50%">Small-strain transversely material with anisotropic damage mechanics using four softening laws. Unrotated axial direction along the y axis.</td>
<td align="center"></td><td align="center">X</td><td align="center">X</td><td align="center">X</td><td align="center">X</td>
</tr>

<tr>
<td><a href="#isoplastsoft"><code>IsoPlasticSoftening</code></a></td>
<td>53</td>
<td width="50%">Small-strain isotropic, plastic material with anisotropic damage mechanics using mode I and mode I softening laws.</td>
<td align="center"></td><td align="center"></td><td align="center">X</td><td align="center">X</td><td align="center">X</td>
</tr>

</table>

<h3>Hyperelastic-Plastic Materials</h3>

<table border="1" cellpadding="2" cellspacing="0" width="90%" align="center">
<tr><th rowspan="2">Name</th><th rowspan="2">Type</th><th rowspan="2">Description</th><th rowspan="2">FEA</th><th colspan="3">2D MPM</th><th rowspan="2">3D MPM</th>
</tr>
<tr><th>P&sigma;</th><th>P&epsilon;</th><th>AS</th>
</tr>

<tr>
<td><a href="#heiso"><code>HEIsotropic</code></a></td>
<td>24</td>
<td width="50%">Isotropic, hyperelastic-plastic material.</td>
<td align="center"></td><td align="center"></td><td align="center">X</td><td align="center">X</td><td align="center">X</td>
</tr>

<tr>
<td><a href="#MGEOS"><code>HEMGEOSMaterial</code></a></td>
<td>25 or 17</td>
<td width="50%">Isotropic, hyperelastic-plastic material using the Mie-Gr&#252;neisen equation of state and a hardening law.</td>
<td align="center"></td><td align="center"></td><td align="center">X</td><td align="center">X</td><td align="center">X</td>
</tr>

<tr>
<td><a href="#DIZ"><code>ClampedNeohookean</code></a></td>
<td>29</td>
<td width="50%">Isotropic, hyperelastic-plastic material with elongations limited to maximum strains</td>
<td align="center"></td><td align="center"></td><td align="center">X</td><td align="center">X</td><td align="center">X</td>
</tr>

</table>

<h3>Viscoelastic, Small Strain Materials</h3>

<table border="1" cellpadding="2" cellspacing="0" width="90%" align="center">
<tr><th rowspan="2">Name</th><th rowspan="2">Type</th><th rowspan="2">Description</th><th rowspan="2">FEA</th><th colspan="3">2D MPM</th><th rowspan="2">3D MPM</th>
</tr>
<tr><th>P&sigma;</th><th>P&epsilon;</th><th>AS</th>
</tr>

<tr>
<td><a href="#visc"><code>Viscoelastic</code></a></td>
<td>6</td>
<td width="50%">Linear viscoelastic material with sum of relaxation times</td>
<td align="center"></td><td align="center">X</td><td align="center">X</td><td align="center">X</td><td align="center">X</td>
</tr>

</table>

<h3>Specialty Materials</h3>

<table border="1" cellpadding="2" cellspacing="0" width="90%" align="center">
<tr><th rowspan="2">Name</th><th rowspan="2">Type</th><th rowspan="2">Description</th><th rowspan="2">FEA</th><th colspan="3">2D MPM</th><th rowspan="2">3D MPM</th>
</tr>
<tr><th>P&sigma;</th><th>P&epsilon;</th><th>AS</th>
</tr>

<tr>
<td><a href="#interface"><code>Interface</code></a></td>
<td>5</td>
<td width="50%">A definition of an imperfect interface</td>
<td align="center">X</td><td align="center"></td><td align="center"></td><td align="center"></td><td align="center"></td>
</tr>

<tr>
<td><a href="#rigid"><code>Rigid</code></a></td>
<td>11</td>
<td width="50%">Either a rigid material (infinite mass and stiffness) or a method to apply moving boundary conditions.</td>
<td align="center"></td><td align="center">X</td><td align="center">X</td><td align="center">X</td><td align="center">X</td>
</tr>

</table>

<h3><a name="tracionlaws"></a>Traction Law Materials</h3>

<table border="1" cellpadding="2" cellspacing="0" width="90%" align="center">
<tr><th rowspan="2">Name</th><th rowspan="2">Type</th><th rowspan="2">Description</th><th rowspan="2">FEA</th><th colspan="3">2D MPM</th><th rowspan="2">3D MPM</th>
</tr>
<tr><th>P&sigma;</th><th>P&epsilon;</th><th>AS</th>
</tr>

<tr>
<td><a href="#traction"><code>TriangularTraction</code></a></td>
<td>12</td>
<td width="50%">A triangular traction law for use on crack surfaces</td>
<td align="center"></td><td align="center">X</td><td align="center">X</td><td align="center">X</td><td align="center"></td>
</tr>

<tr>
<td><a href="#lineart"><code>LinearTraction</code></a></td>
<td>13</td>
<td width="50%">A linear elastic traction law for use on crack surfaces. This law never fails and therefore only implements crack-surface stiffness.</td>
<td align="center"></td><td align="center">X</td><td align="center">X</td><td align="center">X</td><td align="center"></td>
</tr>

<tr>
<td><a href="#cubic"><code>CubicTraction</code></a></td>
<td>14</td>
<td width="50%">A cubic traction law for use on crack surfaces</td>
<td align="center"></td><td align="center">X</td><td align="center">X</td><td align="center">X</td><td align="center"></td>
</tr>

<tr>
<td><a href="#trilinear"><code>TrilinearTraction</code></a></td>
<td>20</td>
<td width="50%">A trilinear traction law for use on crack surfaces</td>
<td align="center"></td><td align="center">X</td><td align="center">X</td><td align="center">X</td><td align="center"></td>
</tr>

<tr>
<td><a href="#coupled"><code>CoupledSawTooth</code></a></td>
<td>23</td>
<td width="50%">A triangular traction law for use on crack surfaces based on a coupled displacement method.</td>
<td align="center"></td><td align="center">X</td><td align="center">X</td><td align="center">X</td><td align="center"></td>
</tr>

<tr>
<td><a href="#plaw"><code>PressureTraction</code></a></td>
<td>26</td>
<td width="50%">A constant stress traction law for creating pressure loaded-cracks.</td>
<td align="center"></td><td align="center">X</td><td align="center">X</td><td align="center">X</td><td align="center"></td>
</tr>

</table>

<h3><a name="contactlaws"></a>Contact Laws</h3>

<p>These contact laws are used to model multimaterial mode contact and crack surface contact. They can model either frictional contact or an imperfect interface.</p>

<table border="1" cellpadding="2" cellspacing="0" width="90%" align="center">
<tr><th>Name</th><th>Number</th><th>Description</th>
</tr>

<tr>
<td><a href="#ignore"><code>IgnoreContact</code></a></td>
<td>60</td>
<td>Ignore contact or revert to single material mode</td>
</tr>

<tr>
<td><a href="#coulomb"><code>CoulombFriction</code></a></td>
<td>61</td>
<td>Contact by Coulomb friction</td>
</tr>

<tr>
<td><a href="#adhesive"><code>AdhesiveFriction</code></a></td>
<td>63</td>
<td>Contact by Coulomb friction with adhesion</td>
</tr>

<tr>
<td><a href="#liquid"><code>LiquidContact</code></a></td>
<td>64</td>
<td>From slip to stick contact for fluid flow</td>
</tr>

<tr>
<td><a href="#linearimp"><code>LinearInterface</code></a></td>
<td>62</td>
<td>An imperfect interface with linear traction laws</td>
</tr>

<tr>
<td><a href="#nonlinearimp"><code>NonlinearInterface</code></a></td>
<td>65</td>
<td>An imperfect interface with potential for nonlinear traction laws</td>
</tr>

</table>

<h3><a name="hardening"></a>Hardening Laws</h3>

<p>These hardening laws can be added to several plasticity materials and the hardening law properties are defined within that material's definition block. Click any one to see the extra the properties needed for that law.</p>

<table border="1" cellpadding="2" cellspacing="0" width="90%" align="center">
<tr><th>Name</th><th>Number</th><th>Description</th>
</tr>

<tr>
<td><a href="#linear"><code>Linear</code></a></td>
<td>1</td>
<td>Linear hardening</td>
</tr>

<tr>
<td><a href="#nonlinear"><code>Nonlinear</code></a></td>
<td>2</td>
<td>Nonlinear hardening</td>
</tr>

<tr>
<td><a href="#nonlinear2"><code>Nonlinear2</code></a></td>
<td>6</td>
<td>Alternate Nonlinear hardening</td>
</tr>

<tr>
<td><a href="#johnsoncook"><code>JohnsonCook</code></a></td>
<td>3</td>
<td>Johnson Cook hardening law</td>
</tr>

<tr>
<td><a href="#scgl"><code>SCGL</code></a></td>
<td>4</td>
<td>Stenberg-Cochran-Guinan hardening</td>
</tr>

<tr>
<td><a href="#sl"><code>SL</code></a></td>
<td>5</td>
<td>Steinberg-Lund hardening</td>
</tr>
</table>

<h3><a name="initiation"></a>Damage Initiation Laws</h3>

<p>These damage initiation laws can be added to <a href="#SSSM">softening materials</a> and the damage initiation law properties are defined within that material's definition block. Click any one to see the extra properties needed for that law.</p>

<table border="1" cellpadding="2" cellspacing="0" width="90%" align="center">
<tr><th>Name</th><th>Number</th><th>Description</th>
</tr>

<tr>
<td><a href="#maxPS"><code>MaxPrinciple</code></a></td>
<td>1</td>
<td>Maximum principle stress or maximum shear stress</td>
</tr>

<tr>
<td><a href="#tiFailure"><code>TIFailure</code></a></td>
<td>1</td>
<td>Damage initiation law specfic for various failure modes in transverely isotropic materials.</td>
</tr>

</table>

<h3><a name="softening"></a>Softening Laws</h3>

<p>These softening laws can be added to <a href="#SSSM">softening materials</a> and the softening law properties are defined within that material's definition block. Click any one to see the extra properties needed for that law.</p>

<table border="1" cellpadding="2" cellspacing="0" width="90%" align="center">
<tr><th>Name</th><th>Number</th><th>Description</th>
</tr>

<tr>
<td><a href="#linearSoft"><code>Linear</code></a></td>
<td>1</td>
<td>Linear softening law</td>
</tr>

<tr>
<td><a href="#expSoft"><code>Exponential</code></a></td>
<td>2</td>
<td>Exponential softening law</td>
</tr>

<tr>
<td><a href="#cubicStep"><code>CubicStep</code></a></td>
<td>3</td>
<td>Cubic step function softening law</td>
</tr>

</table>

<!-- Material  Section -->

<h2><a name="iso"></a><a name="isotropic"></a>Linear Elastic, Isotropic (Type=1)</h2>

<p>This material is small-strain, isotropic material. The material properties are:</p>

<ul>
<li>E - modulus (in <a href="commands.html#units">pressure units</a>)</li>
<li>nu - Poisson's ratio</li>
<li>alpha - thermal expansion coefficient (in ppm/K)</li>
<li>(other) - see the <a href="#common">properties common to all materials</a> (MPM only).</li>
</ul>

<p>|<a href="#tables">Material Tables</a>|</p>
    
<!-- Material  Section -->

<h2><a name="trans1"></a>Linear Elastic, Transversely Isotropic 1 (Type=2)</h2>

<p>This material is small-strain, transversely isotropic with the isotropic plane in the <code>x-y</code> plane. The properties are:</p>

<ul>
<li>EA or ET - axial or transverse modulus (in <a href="commands.html#units">pressure units</a>)</li>
<li>GA - axial shear modulus (in <a href="commands.html#units">pressure units</a>)</li>
<li>nuA or nuT - axial or transverse Poisson's ratio</li>
<li>alphaA or alphaT - axial or transverse thermal expansion coefficient (in ppm/K)</li>
<li>betaA or betaT - axial or transverse concentration expansion coefficient (in strain/wt fraction) (MPM only)</li>
<li>DA or DT - axial or transverse diffusion constant (in <a href="commands.html#units">diffusion units</a>) (MPM only)</li>
<li>kCondA or kCondT - axial or transverse thermal conductivity (in <a href="commands.html#units">conductivity units</a>) (MPM only)</li>
<li>(other) - see the <a href="#common">properties common to all materials</a> (MPM only).</li>
</ul>

<p>|<a href="#tables">Material Tables</a>|</p>
  
<!-- Material  Section -->

<a name="trans2"></a><h2>Linear Elastic, Transversely Isotropic 2 (Type=3)</h2>

<p>This material is small-strain, transversely isotropic with the axial direction of the material in the <code>y</code> direction of the <code>x-y</code> plane analysis. This direction can be rotated by using input <code>angle</code> attributes when assigning material properties to material points or to FEA elements. The input properties are identical to the previous <a href="#trans1">transversely isotropic material</a>.</p>

<p>|<a href="#tables">Material Tables</a>|</p>

<!-- Material  Section -->

<a name="ortho"></a><h2>Linear Elastic, Orthotropic (Type=4)</h2>

<p>The material is small-strain, orthotropic with the input axes corresponding to the analysis <code>x-y-z</code> axes. The material properties can be changed by rotation about the axes:</p>

<ul>
<li>Ex, Ey, or ET - x, y, or z direction modulus (in <a href="commands.html#units">pressure units</a>)</li>
<li>Gxy, Gxz, or Gyz -x-y, x-z, or y-z plane shear modulus (in <a href="commands.html#units">pressure units</a>)</li>
<li>nuxy, nuyx, nuxz, nuzx, nuyz, or nuzy - various Poisson's ratios (enter one of each pair with matching axes)</li>
<li>alphaz, alphay, or alphaz - x, y, or z direction thermal expansion coefficient (in ppm/K)</li>
<li>betaz, betay, or or betaz - x, y, or z direction concentration expansion coefficient (in strain/wt fraction) (MPM only)</li>
<li>Dx, Dy, or Dz - x, y, or z direction diffusion constant (in <a href="commands.html#units">diffusion units</a>) (MPM only)</li>
<li>kCondx, kCOndy, or kCondz - x, y, or z direction thermal conductivity (in <a href="commands.html#units">conductivity units</a>) (MPM only)</li>
<li>(other) - see the <a href="#common">properties common to all materials</a> (MPM only).</li>
</ul>
  
<p>|<a href="#tables">Material Tables</a>|</p>

<!-- Material  Section -->

<a name="interface"></a><h2>Imperfect Interface Material (Type=5)</h2>

<p>This material is used to model imperfect interfaces in FEA.  The input properties are</p>

<ul>
<li>Dn - stiffness for loading the interface in the normal direction (in <a href="commands.html#units">pressure units/length units</a>)</li>
<li>Dt - stiffness for loading the interface in the tangential direction (in <a href="commands.html#units">pressure units</a>/length units)</li>
</ul>

<p>|<a href="#tables">Material Tables</a>|</p>

<!-- Material  Section -->

<h2><a name="visc"></a>Linear Viscoelastic, Isotropic (Type=6)</h2>

<p>This material is a small-strain, linear viscoelastic material with time-dependent shear modulus given by a sum or relaxation times and a time-independent pressure-volume response. The pressure law can use small strain methods (with bulk modulus) or large-deformation <a href="#MGEOS">Mie-Gr&#252;neisen equation of state</a> (MGEOS). The input properties are:</p>

<ul>
<li>pressureLaw - set to 0 to use small strain method (with provided K) or 1 to use MGEOS (with all its properties)</li>
<li>K - the time-independent bulk modulus (in <a href="commands.html#units">pressure units</a>) (when pressureLaw=0)</li>
<li>C0 - the bulk wave speed in MGEOS (in <a href="commands.html#units">alt velocity units</a>) (when pressureLaw=1)</li>
<li>S1 - a parameter in MGEOS (dimensionless) (when pressureLaw=1)</li>
<li>S2 - a parameter in MGEOS (dimensionless) (when pressureLaw=1)</li>
<li>S3 - a parameter in MGEOS (dimensionless) (when pressureLaw=1)</li>
<li>gamma0 - a parameter in MGEOS (dimensionless) (when pressureLaw=1)</li>
<li>Kmax - maximum increase allowed in bulk modulus (by ratio and dimensionless) (default -1 or no max).</li>
<li>G0 - the steady-state (or long time) shear modulus (in <a href="commands.html#units">pressure units</a>)</li>
<li>Gk - the shear modulus for the next relaxation time (in <a href="commands.html#units">pressure units</a>)</li>
<li>tauk - the next relaxation time (in <a href="commands.html#units">time units</a>)</li>
<li>alpha - the thermal expansion coefficient (in ppm/K) (ignored for MGEOS when pressureLaw=1)</li>
<li>(other) - see the <a href="#common">properties common to all materials</a> (MPM only).</li>
</ul>

<p>The relaxation shear modulus is given by G(t) = G<sub>0</sub> + G<sub>1</sub>exp(-t/tau<sub>1</sub>) + G<sub>2</sub>exp(-t/tau<sub>2</sub>) + ... +
G<sub>ntaus</sub>exp(-t/tau<sub>n</sub>), where n is the number of relation times entered (and that number must exactly match the number of Gk shear moduli entered).</p>

<p>This material can be used in plane stress analysis, but only if <code>pressureLaw=0</code> and <a href="#cmn5">artificial visocity</a> is not used.</p>

<h3>History Data for Linear Viscoelastic, Isotropic Materials</h3>

<p>This material tracks history for internal variables needed for implementation of linear viscoelastic properties, but currently those history variables are available for archiving. You can archive total volume change (J) and residual free expansion volume (Jres) in history variables 1 and 2 (Jres, however, is only tracked when using MGEOS).</p>
    
<p>|<a href="#tables">Material Tables</a>|</p>

<!-- Material  Section -->

<h2><a name="mooney"></a>Mooney-Rivlin Hyperelastic, Isotropic (Type=8)</h2>

<p>The material type is a large-deformation, isotropic, hyperelastic material. The material properties are</p>

<ul>
<li>G1 or G2 - the Mooney-Rivlin shear energy depends on two shear moduli (in <a href="commands.html#units">pressure units</a>). Their sum in the low-strain shear modulus.
If G2=0 (which is its default), the material is a Neohookean material.</li>
<li>K - the K modulus (in <a href="commands.html#units">pressure units</a>), which is equal to the low-strain bulk modulus.</li>
<li>E an nu - alternative enter E (in <a href="commands.html#units">pressure units</a>) and nu. These are converted to K and G=G1+G2; you can pick G2 (if not zero) to get G1=G-G2</li>
<li>UJOption - selects the dilational energy term U(J) and the possible integer values mean:<br />
	&#160;&#160;&#160;&#160;0: U(J) = (K/2)((1/2)(J<sup>2</sup>-1) - ln J) (the default)<br />
	&#160;&#160;&#160;&#160;1: U(J) = (K/2)(J-1)<sup>2</sup><br />
	&#160;&#160;&#160;&#160;2: U(J) = (K/2)(ln J)<sup>2</sup><br />
</li>
<li>alpha - the thermal expansion coefficient (in ppm/K)</li>
<li>IdealRubber -  this property should by 0 or 1 where 1 makes the material an ideal rubber elastic material.</li>
<li>(other) - see the <a href="#common">properties common to all materials</a> (MPM only).</li>
</ul>

<h3>History Data for Mooney-Rivlin Materials</h3>

<ol>
<li>The volumetric ratio or J = V/V<sub>0</sub> (<i>i.e.</i>, the determinant of the deformation gradient)</li>
</ol>

<p>The plastic strain stores the left Cauchy Green tensor.</p>

<p>|<a href="#tables">Material Tables</a>|</p>

<!-- New Material Section -->
	
<h2><a name="neoh"></a>Neo-Hookean Material</h2>
	
<p>The material type is an isotropic, hyperelastic material. The material properties are:</p>

<ul>
<li>K, G, Lame, E, and nu - these are the low-strain elastic properties for the material.
You must enter K and G, E and nu, or Lame and G  (moduli in <a href="commands.html#units">pressure units</a>).</li>
<li><code>UJOption</code> for an integer to select the an energy term U(J) and the possible options are:<br />
&#160;&#160;&#160;&#160;0: U(J) = (Lame/2)((1/2)(J<sup>2</sup>-1) - ln J) (the default)<br />
&#160;&#160;&#160;&#160;1: U(J) = (Lame/2)(J-1)<sup>2</sup><br />
&#160;&#160;&#160;&#160;2: U(J) = (Lame/2)(ln J)<sup>2</sup><br />
</li>
<li><code>alpha</code> for thermal expansion coefficient in ppm/K.</li>
<li>(other) - see the <a href="#common">properties common to all materials</a> (MPM only).</li>
</ul>

<p>From the first five properties, you must enter K and G, Lame and G, OR E and nu.</p>

<h3>History Data for Neo-Hookean Material</h3>

<ol>
<li>The volumetric ratio or J = V/V<sub>0</sub> (<i>i.e.</i>, the determinant of the deformation gradient)</li>
</ol>

<p>The plastic strain stores the left Cauchy Green tensor.</p>

<p>|<a href="#tables">Material Tables</a>|</p>

<h2><a name="DIZ"></a>Clamped Neo-Hookean Material (Type=29)</h2>
	
<p>The material type is a large-deformation, isotropic, hyperelastic material. It yields when the tensile and/or compression elongation reaches critical values of 1+&theta;<sub>s</sub> and/or 1-&theta;<sub>c</sub>. The underlying elastic component of the deformation is a neo-Hookean material. After yielding, the mechanical properties soften in tension or harden in compression according to an entered hardening coefficient. The material properties are</p>
	
<ul>
<li>CritTens and CritComp - enter critical strains &theta;<sub>s</sub> and &theta;<sub>c</sub> for critical strains for tension and compression. If either is less than zero, clamping is prevented and the material will use the choosen hyperelastic, neo-Hookean model (see "Elastic" property).</li>
<li>xihard - hardening parameter &xi;. The neo-Hookean moduli are scaled by exp(&xi;(1-J<sub>P</sub>)) where J<sub>P</sub> is the plastic volume ratio.</li>
<li>Elastic - enter 0 to use a modified, corotated, neo-Hookean potential energy defined in paper that proposed this material or enter 1 to use <a href="#neoh">Neo-Hookean material</a> potential energy. This potential energy applies to elastic component of the deformation.</li>
<li>(other) - enter properties for the underlying <a href="#neoh">Neo-Hookean material</a> (note that the UJOption can only be changed when Elastic is 1; UJOption is always 1 when Elastic is 0).</li>
</ul>
	
<h3>History Data for Clamped Neo-Hookean Material</h3>
	
<ol>
	<li>The volumetric ratio or J = V/V<sub>0</sub> (<i>i.e.</i>, the determinant of the deformation gradient)</li>
	<li>The plastic volumetric ratio or J<sub>P</sub> = J/J<sub>E</sub> where J<sub>E</sub> is determinant of the elastic deformation gradient)</li>
</ol>

<p>The plastic strain stores the left Cauchy Green tensor. You have to archive both strain and plastic strain for the visualization to be able to plot elastic and plastic strains.</p>

<!-- Material  Section -->

<p>|<a href="#tables">Material Tables</a>|</p>

<h2><a name="isoplast"></a>Isotropic, Elastic-Plastic Material (Type=9)</h2>

<p> This material is small-strain, isotropic material with plasticity. The new properties are:</p>

<ul>
<li>Hardening - this property selects the <a href="#hardening">hardening law</a>. It should be followed by entering all yielding properties required by the hardening law within the material's definition block.</li>
<li>(other) - see <a href="#iso">Linear Elastic, Isotopic</a> material for all other properties.</li>
</ul>
    
<h3>History Data for Isotropic, Elastic-Plastic Material</h3>

<p>The material has no history variables, but all <a href="#hardening">hardening law</a> will have one or more history variables.</p>

<p>The plastic strain will have the Biot plastic strain. You have to archive both strain and plastic strain for the visualization to be able to plot elastic and plastic strains.</p>

<p>|<a href="#tables">Material Tables</a>|</p>

<!-- Material  Section -->

<h2><a name="bistab"></a>Bistable, Isotropic Elastic (Type=10)</h2>

<p>This material has two small-strain, isotopic states. You need to enter properties of the initial and deformed states and pick rule for state transformations:</p>

<ul>
<li>K0 and Kd - bulk modulus of initial state and transformed state (in <a href="commands.html#units">pressure units</a>).</li>
<li>G0 and Gd - shear modulus of initial state and transformed state (in <a href="commands.html#units">pressure units</a>).</li>
<li>alpha0 and alphad - thermal expansion coefficient of initial state and transformed state (in ppm/K).</li>
<li>beta0 and betad - concentration expansion coefficient of initial state and transformed state (in strain/(wt fraction)).</li>
<li>D0 and Dd - diffusion constant of initial state and transformed state (in <a href="commands.html#units">diffusion units</a>).</li>
<li>kCond0 and kCondd - thermal conductivity of initial state and transformed state (in <a href="commands.html#units">conductivity units</a>).</li>
<li>transition - set state transition to be by a dilation rule (if 1), a distortion rule (if 2), or Von Mises stress rule (if 3) (the default is dilation).</li>
<li>critical - critical value for the transition. The value is volumetric or deviatoric strain (in % for dilation or distortion rule) or stress (in <a href="commands.html#units">pressure units</a>) for Von Mises stress rule.</li>
<li>DeltaVOffset - offset volumetric strain in the transformed state. This property only applies for dilation rule (see above); it is ignored for distortion rule.</li>
<li>reversible - set to "yes" (or 1) to make the transition reversible and "no" (or 0) to make it irreversible (the default is reversible).</li>
<li>(other) - see the <a href="#common">properties common to all materials</a> (MPM only).</li>
</ul>

<h3>History Data for Bistable, Isotropic Elastic Materials</h3>

<ol>
<li>Will be 0 for the initial state and 1 for the deformed state after a transition.</li>
</ol>

<p>|<a href="#tables">Material Tables</a>|</p>
    
<!-- Material  Section -->

<h2><a name="rigid"></a>Rigid Material (Type=11)</h2>

<p>The material is actually two materials in one, depending on the value for its <code>SetDirection</code> property. If <code>SetDirection</code> is 0 through 7, the material is not really a material. Instead material points with such a material will assign grid-based boundary conditions that translate with the particle. If <code>SetDirection</code> is 8, the material will correspond to actual material points that interact with other material points only through multimaterial contact laws. The material properties are:</p>

<ul>

<li>direction - values 0 through 7 (binary 0 through 0x111) create dynamic velocity boundary conditions in the x, y, and/or z axis according to three bits in the binary value (<i>e.g.</i>,1 or binary 0x001 to set velocity in the x direction). A setting of 8 creates rigid particles that can implement multimaterial contact contact.</li>
<li>temperature - set to 1 to create dynamic temperature boundary conditions (it requires direction 0 to 7).</li>
<li>concentration - set to 1 to create dynamic concentration boundary conditions (it requires direction 0 to 7).</li>
<li>SettingFunction, SettingFunction2, or SettingFunction3 - by providing <a href="commands.html#function">user-defined functions</a> of time, position, and time step, the rigid particle velocity setting can vary. The functions should evaluate to the velocity in <a href="commands.html#units">velocity units</a>. For SetDirection 1 to 7 the functions must be given in order and apply to the set velocities in order. For SetDirection of 8, the three functions are for setting <code>x</code>, <code>y</code>, and <code>z</code> directions and zero to three can be used.</li>
<li>SettingFunctionX, SettingFunctionY, or SettingFunctionY - synonyms for SettingFunction, SettingFunction2, or SettingFunction3, respectively.</li> 
<li>ValueFunction - by entering <a href="commands.html#function">user-defined functions</a> of time, time step, and position, the rigid particle temperature and concentration can vary. The function should evaluate to the temperature (in K) or concentration potential.</li>
<li><code>mirrored</code> should be set to -1 or +1 to indicate that rigid particles with SetDirection = 1 to 7 are setting a velocity on the minimum edge (-1) or maximum edge (+1) of the object along the moving direction. This property can improve the performance of velocity boundary conditions as explained on the <a href="http://osupdocs.forestry.oregonstate.edu/index.php/Rigid_Material#Mirrored_Velocity_Property">OSUPdocs wiki</a>. The default setting of 0 means to not use mirroring. Note that when <a href="commands.html#mhother">ExtrapolateRigid</a> option is activated, this property is ignored.</li>

</ul>

<p>|<a href="#tables">Material Tables</a>|</p>

<!-- Material  Section -->

<h2><a name="hill"></a>Hill Plastic, Orthotropic Material (Type=15)</h2>

<p>This material is a small-strain, elastic-plastic material based on an <a href="#ortho">orthotropic</a> material but has plasticity determined by a built-in, anisotropic criterion for yielding:</p>
<blockquote>[ F(&sigma;<sub>yy</sub> - &sigma;<sub>zz</sub>)<sup>2</sup> + G(&sigma;<sub>xx</sub> - &sigma;<sub>zz</sub>)<sup>2</sup>
    + H(&sigma;<sub>yy</sub> - &sigma;<sub>xx</sub>)<sup>2</sup> + 2L&tau;<sub>yz</sub><sup>2</sup>
    + 2M&tau;<sub>xz</sub><sup>2</sup> + 2N&tau;<sub>xy</sub><sup>2</sup> ]<sup>1/2</sup> - (1 + K&epsilon;<sub>p</sub><sup>n</sup>)
</blockquote>
<p>where K and n are dimensionless hardening law properties and F through N are determined from yield stress in each material direction. The material properties are:
</p>

<ul>

<li>yldxx - yield stress from uniaxial tension test the material's <code>x</code> direction (in <a href="commands.html#units">pressure units</a>)</li>
<li>yldyy - yield stress from uniaxial tension test the material's <code>y</code> direction (in <a href="commands.html#units">pressure units</a>)</li>
<li>yldzz - yield stress from uniaxial tension test the material's <code>z</code> direction (in <a href="commands.html#units">pressure units</a>)</li>
<li>yldxy - yield stress from pure shear test in the material's <code>x-y</code> plane (in <a href="commands.html#units">pressure units</a>)</li>
<li>yldyz - yield stress from pure shear test in the material's <code>y-z</code> plane (in <a href="commands.html#units">pressure units</a>)</li>
<li>yldxz - yield stress from pure shear test in the material's <code>x-z</code> plane (in <a href="commands.html#units">pressure units</a>)</li>
<li>khard - hardening term (see above)</li>
<li>nhard - exponent on cumulative plastic strain in hardening term (see above)</li>
<li>(other) - see <a href="#ortho">Linear Elastic, Orthotropic</a> material for all other properties.</li>
</ul>
<h3>History Data for Hill Plastic, Orthotropic Material</h3>

<ol>
<li>The cumulative equivalent plastic strain defined as the sum of
<code>sqrt(2/3)||d&epsilon;<sup>p</sup>||</code>.</li>
</ol>

<p>The plastic strain will have the plastic strain. You have to archive both strain and plastic strain for the visualization to be able to plot elastic and plastic strains.</p>

<p>|<a href="#tables">Material Tables</a>|</p>
    
<!-- Material  Section -->

<h2><a name="MGEOS"></a>Mie-Gr&#252;neisen Equation of State Materials (Type=25 or Type=17)</h2>

<p>This material is isotropic with plasticity and find pressure using the Mie-Gr&#252;neisen equation of state by large deformation methods. It is a large-deformation, hyperelastic-plastic methods for shear and yielding can use any <a href="#hardening">hardening law</a> for plasticity. Note: the Type=17 material used to be a small-strain material, but is now deleted and both types now use large deformation methods. The material properties are:
</p>

<ul>
<li>C0 - the bulk wave speed (in <a href="commands.html#units">alt velocity units</a>).</li>
<li>S1 - a parameter in Mie-Gr&#252;neisen equation of state (dimensionless).</li>
<li>S2 - a parameter in Mie-Gr&#252;neisen equation of state (dimensionless).</li>
<li>S3 - a parameter in Mie-Gr&#252;neisen equation of state (dimensionless).</li>
<li>gamma0 - a parameter in Mie-Gr&#252;neisen equation of state (dimensionless).</li>
<li>Kmax - maximum increase allowed in bulk modulus (by ratio and dimensionless) (default is -1 or no max).</li>
<li>G - the shear modulus at zero pressure and the reference temperature (in <a href="commands.html#units">pressure units</a>).</li>
<li>Hardening - this property selects the <a href="#hardening">hardening law</a>. It should be followed by entering all yielding properties required by the hardening law within the material's definition block.</li>
<li>(other) - see the <a href="#common">properties common to all materials</a>. This material requires <a href="#common">heat capacity <code>Cv</code></a> and a <a href="commands.html#thermal"><code>StressFreeTemp</code></a> in K.</li>
</ul>

<p>In this material, the pressure is related to the volumetric compression strain (x = -&Delta;V/V<sub>0</sub>) by the Mie-Gr&#252;neisen equation of state. When x>0, the material is in compression and pressure is
</p>
<blockquote>
   P = &rho;<sub>0</sub>C<sub>0</sub><sup>2</sup>x(1 - 0.5 &gamma;<sub>0</sub> x) /
   (1 - S<sub>1</sub>x - S<sub>2</sub>x<sup>2</sup> - S<sub>3</sub>x<sup>3</sup>)<sup>2</sup>
   + &rho;<sub>0</sub>&gamma;<sub>0</sub>e
</blockquote>
<p>where e is the internal energy. When x&lt;0, the material is in tension and the pressure changes to</p>
<blockquote>
   P = &rho;<sub>0</sub>C<sub>0</sub><sup>2</sup>x
   + &rho;<sub>0</sub>&gamma;<sub>0</sub>e
</blockquote>
 <p>The first term uses the low-strain bulk modulus, which is
 &rho;<sub>0</sub>C<sub>0</sub><sup>2</sup>.
</p>

<p>This equation of state automatically handles thermal expansion (assuming a valid Cv was provided) and therefore you do not need to enter a thermal expansion coefficient.</p>

<h3>History Data for Mie-Gr&#252;neisen Equation of State Material</h3>

<p>All <a href="#hardening">hardening law</a> will have one or more history variables. This material then tracks relative volume (J = V/V<sub>0</sub>) in the first history variable after hardening law history variables.</p>

<p>The plastic strain stores the elastic, left Cauchy Green tensor. You have to archive both strain and plastic strain for the visualization to be able to plot elastic and plastic strains.</p>
    
<p>|<a href="#tables">Material Tables</a>|</p>

<!-- Material  Section -->

<h2><a name="heaniso"></a>Anisotropic, Elastic-Plastic, Hyperelastic Material (Type=21)</h2>

<p>The material type is a large-strain, anisotropic, hyperelastic material for fabric processing. It is currently in development (in OSParticulus only).</p>

<p>|<a href="#tables">Material Tables</a>|</p>

<!-- Material  Section -->

<h2><a name="idealgas"></a>Ideal Gas, as a Hyperelastic Material (Type=22)</h2>

<p>The material type is an ideal gas, implemented as a large-deformation, isotropic, hyperelastic material. The possible input properties are:</p>

<ul>
<li>P0 - the reference pressure (in <a href="commands.html#units">pressure units</a>) at reference temperature and reference density.</li>
<li>T0 - the reference temperature (in K)</li>
<li>rho - the reference density (in <a href="commands.html#units">density units</a>) at reference temperature.</li>
<li>Cv - if heat capacity is omitted or is less than or equal 1, C<sub>V</sub> is set to (3/2)R for a monotonic gas. If the entered value is greater than 1, C<sub>V</sub> is set to (5/2)R for a diatomic gas.</li>
<li>(other) - see the <a href="#common">properties common to all materials</a> (MPM only).</li>
</ul>

<p>|<a href="#tables">Material Tables</a>|</p>

<!-- Material  Section -->

<h2><a name="heiso"></a>Isotropic, Hyperelastic-Plastic Material (Type=24)</h2>

<p>The material is a large-deformation, isotropic, hyperelastic-plastic material. The elastic stresses use a Neo-Hookean material model. The properties are:</p>

<ul>
<li>Hardening - this property selects the <a href="#hardening">hardening law</a>. It should be followed by entering all yielding properties required by the hardening law within the material's definition block.</li>
<li>(other) - see <a href="#mooney">Mooney-Rivlin</a> material for all other properties, except this material does not support (or allow) the G2 or the<code> &lt;IdealRubber/&gt;</code> properties.</li>
</ul>


<h3>History Data for Isotropic Hyperelastic-Plastic Materials</h3>

<p>The first history variables are determined by the <a href="#hardening">hardening law</a>. After those variables, the next  history variable stores the volumetric strain J = V/V<sub>0</sub> (<i>i.e.</i>, the determinant of the deformation gradient).</p>

<p>The plastic strain stores the elastic, left Cauchy Green tensor. You have to archive both strain and plastic strain for the visualization to be able to plot elastic and plastic strains.</p>

<p>|<a href="#tables">Material Tables</a>|</p>

<!-- Material  Section -->

<h2><a name="tait"></a>Tait Liquid (Type=27)</h2>

<p>This material models a Newtonian fluid as a hyperelastic material. The pressure is found from the Tait equation. The properties are:</p>

<ul>
<li>K - the zero-pressure, bulk modulus (in <a href="commands.html#units">pressure units</a>)</li>
<li>viscosity - liquid viscosity at one shear rate (in <a href="commands.html#units">viscosity units</a>)</li>
<li>logshearrate - log of shear rate corresponding to a viscosity (in <a href="commands.html#units">1/time units</a>).</li>
<li>alpha - the thermal expansion coefficient (in ppm/K)</li>
<li>InitialPressure - set initial pressure to <a href="commands.html#function">user-defined function</a> of
position that evaluates to a pressure in <a href="commands.html#units">pressure units</a>.
When liquids are modeled in gravity, the function can set to &rho;gh,
where h is height of liquid above the position (x, y, z).</li>
<li>(other) - see the <a href="#common">properties common to all materials</a> (MPM only).</li>
</ul>

<p>If one <code>viscosity</code> is given and no <code>logshearrate</code>, the material has constant viscosity. If multiple <code>logshearrate</code> and <code>viscosity</code> commands are used, each pair defines points for piecewise linear representation of viscosity as a function of log shear rate. You must enter an equal number of <code>logshearrate</code> and <code>viscosity</code> commands with monotonically increasing shear rates. The liquid's viscosity will be interpolated within the provided points. Shear rates below the minimum or above the maximum provided shear rates will be equal to the viscosity at the minimum or maximum shear rate, respectively.</p> 

<h3>History Variables for Tait Liquid</h3>

<p>The following history variables are stored:</p>
<ol>
<li>Volumetric strain ratio J = V/V<sub>0</sub> (<i>i.e.</i>, the determinant of the deformation gradient).</li>
<li>Volumetric residual strain ratio J<sub>res</sub>.</li>
<li>Shear rate.</li>
</ol>

<p>|<a href="#tables">Material Tables</a>|</p>

<!-- Material  Section -->

<h2><a name="isosoft"></a>Isotropic, Small-Strain Material with Anisotropic Damage (Type=50)</h2>

<p>The material is an isotropic, small strain material and currently only in OSParticulas. Once damage initiates (using any available <a href="#initiation">initiation law</a>), the material develops a crack and propagates damage using anisotropic damage mechanics. The damage process is controlled by any available <a href="#softening">softening law</a>. For more details see the <a href="http://osupdocs.forestry.oregonstate.edu/index.php/Isotropic_Softening_Material">OSUPDocs wiki</a>.The properties are:</p>

<ul>
<li>Initiation - this property selects the <a href="#initiation">damage initiation law</a>. It should be followed by entering all properties required by the damage initiation law within the material's definition block. The default is <a href="#maxPS"><code>MaxPrinciple</code></a>.</li>
<li>SofteningI - this property selects the mode I <a href="#softening">softening law</a>. It should be followed by entering all properties required by the softening law within the material's definition block. Each property should be prefaced with "I-". For example, to enter mode I law toughness for "Gc", enter it ias "I-Gc". The default is <a href="#linearSoft">linear softening law</a>.</li>
<li>SofteningII - this property is same as "SofteningI" except it selects the mode II <a href="#softening">softening law</a> and properties for this law should be prefaced with "II-".</li>
<li>(other) - see <a href="#iso">Isotropic</a> material for all properties of the underlying small-strain, isotropic material, except this softening material requires use of <a href="#cmn1"><code>largeRotation</code> property</a> set to 1.</li>
</ul>

<p>The particle plastic strain will archive the total cracking strain in the global axis system. The plastic energy will archive the total amount of energy released by damage to the particle.</p>

<h3>History Data for Isotropic, Small-Strain Materials with Anisotropic Damage</h3>

<ol>
<li>0, 0.9, 1.1, 1.9, or 2.1 to indicate undamaged (0), damage propagation (0.9 or 1.1), or post failure (decohesion) state of the particle (1.9 or 2.1). 0.9 and 1.9 indicate the failure initiated by tensile strength while 1.1 and 2.1 indicate failure initiated by shear strength.</li>
<li>The maximum normal cracking strain.</li>
<li>The maximum x-y shear cracking strain.</li>
<li>The maximum x-z cracking strain (zero for 2D).</li>
<li>d<sub>n</sub> or damage variable for normal loading.</li>
<li>d<sub>xy</sub> or damage variable for x-y shear loading.</li>
<li>d<sub>xz</sub> or damage variable for x-z shear loading (zero for 2D).</li>
<li>For 2D it is cos(&alpha;), but for 3D it is Euler angle &alpha;.</li>
<li>For 2D it is sin(&alpha;), but for 3D it is Euler angle  &beta;.</li>
<li>For 2D it is not used, but for 3D it is Euler angle &gamma;.</li>
<li>A<sub>c</sub>/V<sub>p</sub> where A<sub>c</sub> is crack area within the particle and V<sub>p</sub> is particle volume.</li>
<li>Relative strength derived at the start by <code>strengthCoefVariation</code> property.</li>
</ol>

<p>History variables 8-10 are storing Euler angles &alpha;, &beta;, and &gamma; (in radians) for a Z-Y-Z rotation scheme; only &alpha; is needed for 2D and it is stored as cos(&alpha;) and sin(&alpha;) instead of as angles.</p>

<p>|<a href="#tables">Material Tables</a>|</p>

<!-- Material  Section -->

<h2><a name="transisosoft"></a>Transversely isotropic, Small-Strain Material with Anisotropic Damage 1 and 2 (Types=51 and 52)</h2>

<p>The material is an transversely isotropic, small strain material and currently only in OSParticulas. Once damage initiates (using any appropriate <a href="#initiation">initiation law</a>), the material develops a crack and propagates damage using anisotropic damage mechanics. The damage process is controlled by available <a href="#softening">softening laws</a>. Type 1 has the unique axis in the <code>z</code> direction; this material is isotropic in the <code>x-y</code> plane when running 2D analyses. Type 2 has the unique axis in the <code>y</code> direction. Any other orientation is achieved by rotating from these initial orientations. For more details see the <a href="http://osupdocs.forestry.oregonstate.edu/index.php/Transversely_Isotropic_Softening_Material">OSUPDocs wiki</a>.The properties are:</p>

<ul>
<li>Initiation - this property selects a <a href="#initiation">damage initiation law</a> appropriate for transversely isotropic materials. It should be followed by entering all properties required by the damage initiation law within the material's definition block. The default is <a href="#tiFailure"><code>TIFailure</code></a>.</li>
<li>SofteningEA</li>
<li>SofteningGA</li>
<li>SofteningET</li>
<li>SofteningGT - these four properties select the <a href="#softening">softening law</a> for propagation of damage that changes EA, GA, ET, and GT, respectively. After one is attached (and all four are needed), it should be followed by properties required by the softening law within the material's definition by prefacing the property "EA-", "GA-", "ET-", or "GT-", respectively. The default is <a href="#linearSoft">linear softening law</a>.</li>
<li>shearFailureSurface - select failure surface assumed when modeling shear damage in 3D calculations. Use 1 for an elliptical failure criterion or 0 for a rectangular failure surface (default is 1).</li>
<li>strengthCoefVariation - this property assigns a coefficient of variation to all strength properties. Each particle's strength is set at the start of the simulation to have the same Gaussian distribution of values about their means, but will have no spatial correlations.</li>
<li>(other) - see <a href="#isotropic">Isotropic</a> material for all properties of the underlying small-strain, isotropic material, except this softening material requires use of <a href="#cmn1"><code>largeRotation</code> property</a> set to 1.</li>
</ul>

<p>The particle plastic strain will archive the total cracking strain in the global axis system. The plastic energy will archive the total amount of energy released by damage to the particle.</p>

<h3>History Data for Isotropic, Small-Strain Materials with Anisotropic Damage</h3>

<ol>
<li>0, 0.75, 0.8, 0.85, 0.95, 1.05, 1.15, 1.25, or 1 higher than previous 7 to indicate undamaged (0), damage propagation (0.75, 0.8, 0.85, 0.95, 1.05, 1.15, 1.25), or post failure (decohesion) state of the particle.
<ul>
<li>0.75, 0.8, or 0.85 indicate failure by transverse tension (0.75), axial shear (0.8), or rolling shear (0.85) and material axial direction along z axis in the crack axis system</li>
<li>0.95 or 1.05 indicate failure by axial tension (0.95) or transverse shear (1.05) and material axial direction along x axis in the crack axis system</li>
<li>1.15 or 1.25 indicate failure by axial shear (1.15) or transverse tension (1.25) and material axial direction along y axis in the crack axis system</li>
</ul>
</li>
<li>δ<sub>n</sub> or the maximum normal cracking strain.</li>
<li>δ<sub>xy</sub> or the maximum x-y shear cracking strain.</li>
<li>δ<sub>xz</sub> or the maximum x-z cracking strain (zero for 2D).</li>
<li>d<sub>n</sub> or damage variable for normal loading.</li>
<li>d<sub>xy</sub> or damage variable for x-y shear loading.</li>
<li>d<sub>xz</sub> or damage variable for x-z shear loading (zero for 2D).</li>
<li>For 2D it is cos(&alpha;), but for 3D it is Euler angle &alpha;.</li>
<li>For 2D it is sin(&alpha;), but for 3D it is Euler angle  &beta;.</li>
<li>For 2D it is not used, but for 3D it is Euler angle &gamma;.</li>
<li>A<sub>c</sub>/V<sub>p</sub> where A<sub>c</sub> is crack area within the particle and V<sub>p</sub> is particle volume.</li>
<li>Relative strength derived at the start by <code>strengthCoefVariation</code> property.</li>
</ol>

<p>History variables 8-10 are storing Euler angles &alpha;, &beta;, and &gamma; (in radians) for a Z-Y-Z rotation scheme; only &alpha; is needed for 2D and it is stored as cos(&alpha;) and sin(&alpha;) instead of as angles.</p>

<p>|<a href="#tables">Material Tables</a>|</p>

<!-- New Material Section -->

<h2><a name="isoplastsoft"></a>Isotropic, Plastic, Small-Strain Material with Anisotropic Damage (Type=53)</h2>

<p>This material is an isotropic, elastic-plastic material that can also develop aniostropic damage. For more details see the <a href="http://osupdocs.forestry.oregonstate.edu/index.php/Isotropic_ Plastic Softening Material">OSUPDocs wiki</a>.The properties are:

<ul>
<li><a href="#isotropic">(Isotropic, Elastic Properties)</a> - Enter all properties needed to define the underlying isotropic material response</li>
<li><a href="#isoplast">(Isotropic, Plastic Properties)</a> - Enter yield properties and a hardening law (but cannot use a hardening law that changes shear modulus)</li>
<li><a href="#isosoft">(Isotropic, Softening Properties)</a> - Enter properties for initiation of damage and for two softening laws</li>
<li>(other) - see <a href="#isotropic">Isotropic</a> material for all properties of the underlying small-strain, isotropic material, except this softening material requires use of <a href="#cmn1"><code>largeRotation</code> property</a> set to 1.</li>
</ul>

<p>The particle plastic strain will archive the sum of plastic and cracking strain in the global axis system. The plastic energy will archive the total amount of energy released by plsticity or by damage to the particle.</p>

<h3>History Data for Isotropic, Plastic, Small-Strain Materials with Anisotropic Damage</h3>

<ul>
<li>The first history variables for for the choosen softening law.
<li>The subsequent history variables are identical to <a href="#isosoft">isotropic softening materials</a>, but renumbered from end of softening law history variables</li>
<li>Three addition history variable at the end will track components of the normal and shear cracking strains in the crack axis system.</li>
</ul>

<p>|<a href="#tables">Material Tables</a>|</p>

<!-- Material  Section -->

<h2><a name="traction"></a>Triangular Traction Law Material (Type=12)</h2>

<p>The traction law has a triangular shape. There are separate traction laws for opening displacement (mode I) and sliding displacements (mode II):
</p>
<center><img src = "tritraction.jpg" width = "291" height = "185" alt = "triangular traction law">
</center>

<ul>

<li>sigmaI and sigmaII - the peak stress or cohesive stress of the traction law (&sigma;) (in <a href="commands.html#units">pressure units</a>).</li>
<li>JIc and JIIc - the toughness of the traction law or area under the traction law curve (J<sub>c</sub>) (in <a href="commands.html#units">energy release units</a>)</li>
<li>delIc and delIIc - the maximum or critical opening displacement (&delta;<sub>c</sub>) at which debonding occurs (in <a href="commands.html#units">length units</a>)</li>
<li>delpkI and delpkII - the relative displacement at the location of the peak in the triangular law (relative to &delta;<sub>c</sub> and therefore dimensionless from 0 to 1).</li>
<li>kIe and kIIe - the initial elastic slope (k) (in <a href="commands.html#units">pressure units/length units</a>).</li>
<li>nmix - the exponent used in the mixed-mode failure criterion Enter nmix&lt;=0 to choose infinity, which is decoupled failure criterion (see <a href="#mixedmode">below</a>).</li>
</ul>

<p>You must then enter exactly two of the first three properties for each mode (because they are interrelated). You must enter zero or one for the next two properties. If one is provided, the other will be calculated. If neither k nor &delta;<sub>peak</sub> are entered, the peak will be located at &delta;<sub>peak</sub> = 0.225926299&delta;<sub>c</sub>.</p>

<p><a name="mixedmode"></a>Failure occurs when</p>
<blockquote>(G<sub>I</sub>/J<sub>Ic</sub>)<sup>nmix</sup> + (G<sub>II</sub>/J<sub>IIc</sub>)<sup>nmix</sup> = 1
</blockquote>
<p>where G<sub>I</sub> and G<sub>II</sub> are areas under mode I and mode II traction laws up to current displacements. If nmix&lt;=0, failure occurs when either mode reaches its critical COD.
</p>

<p>|<a href="#tables">Material Tables</a>|</p>

<!-- Material  Section -->

<a name="lineart"></a><h2>Linear Elastic Traction Law Material (Type=13)</h2>

<p>This traction law creates a linear elastic traction law that never fails and therefore never releases energy. The only properties are the slopes for normal and shear opening.
</p>
<center><img src = "linear.jpg" width = "291" height = "185" alt = "linear elastic traction law">
</center>


<ul>

<li>kIe and kIIe - the slope of the linear elastic tractional law (k) (in <a href="commands.html#units">pressure units/length units</a>).</li>

</ul>

<p>If either slope is left out, it will be set to zero. To create a linear elastic traction law that fails at some critical cod, use a <a href="#traction">triangular traction law material</a> instead.
</p>

<p>|<a href="#tables">Material Tables</a>|</p>

<!-- Material  Section -->

<a name="cubic"></a><h2>Cubic Traction Law Material (Type=14)</h2>

<p>The traction law uses a cubic function:</p>
<blockquote>
&sigma; = (27/4) &sigma;<sub>max</sub> (&delta;/&delta;<sub>c</sub>)(1-(&delta;/&delta;<sub>c</sub>))<sup>2</sup>.
</blockquote>
<p>that was designed to have zero slope at the critical COD.  There are separate traction laws for opening displacement (mode I) and sliding displacements (mode II):
</p>
<center><img src = "cubictraction.jpg" width = "291" height = "185" alt = "cubic traction law">
</center>

<ul>

<li>sigmaI and sigmaII - the peak stress or cohesive stress of the traction law (&sigma;) (in <a href="commands.html#units">pressure units</a>).</li>
<li>JIc and JIIc - the toughness of the traction law or area under the traction law curve (J<sub>c</sub>) (in <a href="commands.html#units">energy release units</a>).</li>
<li>delIc and delIIc - the maximum or critical opening displacement (&delta;<sub>c</sub>) at which debonding occurs (in <a href="commands.html#units">length units</a>).</li>
<li>nmix - the exponent used in the <a href="#mixedmode">mixed-mode failure criterion</a>. Enter nmix&lt;=0 to choose infinity, which is decoupled failure criterion.</li>
</ul>

<p>You must then enter exactly two of the first three properties for each mode. The failure criterion under possible mixed-mode conditions is identical to the one used in the <a href="#mixedmode">triangular traction law</a>.
</p>

<p>|<a href="#tables">Material Tables</a>|</p>

<!-- Material  Section -->

<a name="trilinear"></a><h2>Trilinear Traction Law Material (Type=20)</h2>

<p>This traction law is a piecewise-linear law with two break points.There are separate traction laws for opening displacement (mode I) and sliding displacements (mode II):</p>
<center><img src = "trilintraction.jpg" width = "291" height = "185" alt = "triangular traction law">
</center>


<ul>
<li>sigmaI, sigmaII, JIc, JIIc, delIc, delIIc, kIe, kIIe, delpkI, delpkII, and nmix - these properties are defined in the <a href="#traction">Triangular Traction Law</a> and they all mean the same thing for this traction law, except that they now apply to the first break point in the trilinear law at (&delta;<sub>1</sub>,&sigma;<sub>1</sub>).</li>
<li>sigmaI2 and sigmaII2 - the stress at the second break point in the traction traction law (&sigma;<sub>2</sub>) (in <a href="commands.html#units">pressure units</a>).</li>
<li>delpkI2 and delpkII2 - the relative displacement at the location of the second peak in the trilinear law (relative to &delta;<sub>c</sub> and therefore dimensionless from 0 to 1).</li>
</ul>

<p>Excluding nmix, you must enter exactly 5 of the seven properties for each mode. Furthermore, if one of the 5 is kIe (or kIIe for mode II), then either sigmaI or delpkI (but not both) must be specified as well.</p>

<p>The failure under mixed-mode loading uses the same criterion explained <a href="#mixedmode">above</a>.</p>

<p>|<a href="#tables">Material Tables</a>|</p>

<!-- Material  Section -->

<a name="coupled"></a><h2>CoupledSawTooth Traction Law Material (Type=23)</h2>

<p>This traction law also has a triangular shape except that cod and traction are now effective terms that are related to normal and tangential components of traction and displacement:</p>
<blockquote>
Traction = <code>T<sub>eff</sub></code> where  <code>T<sub>eff</sub><sup>2</sup> = T<sub>n</sub><sup>2</sup> + T<sub>t</sub><sup>2</sup></code><br>
cod = <code>&delta;<sub>eff</sub></code> where  &delta;<sub>eff</sub><sup>2</sup> = &delta;<sub>n</sub><sup>2</sup> + &delta;<sub>t</sub><sup>2</sup>
</blockquote>

<center><img src = "tritraction.jpg" width = "291" height = "185" alt = "triangular traction law">
</center>


<p>The properties are:</p>

<ul>
<li>sigmaI, JIc, delIc, kIe, and delpkI - these properties are defined in the <a href="#traction">Triangular Traction Law</a> and they all mean the same thing for this traction law, except they apply now to effective traction and critical effective cod rather than mode I traction and critical cod. Mode II parameters and nmix are ignored.</li>
</ul>

<p>|<a href="#tables">Material Tables</a>|</p>

<!-- Material  Section -->

<a name="plaw"></a><h2>Pressure Traction Law Material (Type=26)</h2>

<p>This traction law creates a constant stress that is normal to the crack surface. It models a pressure loaded crack.
The stress can be entered in one of two ways:
</p>

<ul>

<li>stress - enter constant normal stress; use negative stress for a pressure loaded crack (in <a href="commands.html#units">pressure units</a>).</li>
<li>function - Alternatively, the stress can be entered as a <a href="commands.html#function">user-defined function</a> of time (in <a href="commands.html#units">pressure units</a>).</li>
<li>minCOD - specify a minimum, normal, crack opening displacement below which no pressure is applied (in <a href="commands.html#units">length units</a>).
It is ignored if it is negative (default is -1)</li>
</ul>

<p>|<a href="#tables">Material Tables</a>|</p>

<!-- Material  Section -->

<a name="ignore"></a><h2>Contact Law that Ignores Contact (Type=60)</h2>

<p>This law is not actually a contact law. This law ignores contact on explicit crack surfaces. For <a href="commands.html#mmmode">multimaterial contact</a>, this law will revert to a single velocity field as if they was not a material interface. The law has no properties that can be set.</p>

<p>This law will give poor results for cracks that are in contact, unless those cracks never experience contact. For <a href="commands.html#mmmode">multimaterial mode</a> simulations, this mode reverts to a single velocity field. For simulations with more than two materials where some contact by other contact laws and others should use single velocity fields, the better approach than using this law is to use the <a href="#cmn4">shareMatField (matID)</a> in materials that should share the same field.
</p>

<p>|<a href="#tables">Material Tables</a>|</p>

<!-- Material  Section -->

<a name="coulomb"></a><h2>Coulomb Friction Contact Law (Type=61)</h2>

<p>This law implements simple Coulomb friction law where sliding traction is proportional to the contacting nomral traction:
</p>

<blockquote>
<code>S<sub>slide</sub> = µ N</code>
</blockquote>

<p>and µ is the dynamic coefficient of friction. If the friction coefficient is entered as a negative number, then surfaces stick when in contact but move freely when separated. The properties for this law are:</p>

<ul>

<li><code>coeff</code> for the dynamic coefficient of friction (default is 0).</li>
<li><code>coeffStatic</code> for the static coefficient of friction (default is -1).</li>
<li><code>displacementOnly</code> 0 for contact by separation and stress, 1 for contact by separation only, or &lt;0 for contact by sepaction and interfacial stress &lt; <code>-displacementOnly</code> (in <a href="commands.html#units">pressure units/length units</a>) (default is 0).</li>
<li><code>Dc</code> &lt;0 to find separation assuming perfect interfaciall contact, &ge;0 to use <a href="#linearimp">imperfect interface methods</a> to find separation (in <a href="commands.html#units">pressure units/length units</a>) (default is -1)</li>
</ul>

<p>If the optional static coefficient of friction is changed to a positive number, it must be greater than the dynamic coefficient. The sliding will begin when it overcomes the static frictional force, but thereafter will slide with the dynamic coefficient of friction.</p>

<p>|<a href="#tables">Material Tables</a>|</p>

<!-- Material  Section -->

<a name="adhesive"></a><h2>Coulomb Friction with Adhesion Contact Law (Type=63)</h2>

<p>This law implements Coulomb friction law with adhesion (in OSParticulus only). If the surface are in contact,
the frictional sliding force is
</p>

<blockquote>
<code>S<sub>slide</sub> = S<sub>a</sub> + (µ +k&Delta;v)N</code>
</blockquote>

<p>where S<sub>a</sub> is the tangential adhesion strength and µ is the coefficient of friction, and k allows the dynamic coefficient of friction to depend linearly on sliding velocity. The µ term models transition from static to dynamic coefficient of friction by</p>

<blockquote>
<code>µ = (µ<sub>s</sub> vhalf + µ<sub>d</sub> v)/(vhalf + v)</code>
</blockquote>

<p>where <code>vhalf</code> is velocity for which coefficient of friction is half way from static coefficient of friction (µ<sub>s</sub>) to dynamic coefficient of friction (µ<sub>d</sub>). The default is <code>vhalf</code>=0, which corresponds to instantaneous drop from µ<sub>s</sub> to µ<sub>d</sub> at the onset of sliding.</p>

<p>If the surfaces are not in contact, the surface continue to stick as long as:</p>

<blockquote>
<code>(S<sub>stick</sub>/S<sub>a</sub>)<sup>2</sup> + (N/N<sub>a</sub>)<sup>2</sup> &lt; 1</code>
</blockquote>

<p>where N<sub>a</sub> is the normal adhesion strength. If this criterion is not met (or if either Sa or Na are zero), the surfaces move freely with zero tractions. The properties for this law are:</p>

<ul>
<li><code>coeff</code> for the dynamic coefficient of friction (default is 0).</li>
<li><code>coeffStatic</code> for the static coefficient of friction (default is -1).</li>
<li><code>Sa</code> for tangential adhesion strength (in <a href="commands.html#units">pressure units</a>) (default is 0).</li>
<li><code>Na</code> for normal adhesion strength (in <a href="commands.html#units">pressure units</a>) (default is 0).</li>
<li>kmu - slope of coefficient of friction vs. velocity (in <a href="commands.html#units">1/velocity units</a>) (default is 0).</li>
<li>vhalf - velocity at which µ is half way from µ<sub>s</sub> to µ<sub>d</sub> (in <a href="commands.html#units">velocity units</a>) (default is 0).</li>
<li><code>displacementOnly</code> 0 for contact by separation and stress, 1 for contact by separation only, or &lt;0 for contact by sepaction and interfacial stress &lt; <code>-displacementOnly</code> (in <a href="commands.html#units">pressure units/length units</a>) (default is 0).</li>
<li><code>Dc</code> &lt;0 to find separation assuming perfect interfaciall contact, &ge;0 to use <a href="#linearimp">imperfect interface methods</a> to find separation (in <a href="commands.html#units">pressure units/length units</a>) (default is -1)</li>
</ul>

<p>If the optional static coefficient of friction is changed to a positive number,
it must be greater than the dynamic coefficient. The sliding will begin when
the sliding force overcomes the frictional force calculated
from static coefficient and v=0, but thereafter will slide with force
found using the velocity dependent dynamic coefficient of friction.</p>

<p>|<a href="#tables">Material Tables</a>|</p>

<!-- Material  Section -->

<a name="liquid"></a><h2>Liquid Contact Law (Type=64)</h2>

<p>This frictional contact law implements a friction-style law between liquid and a wall where contact shear is related to shear rate, viscosity, and a scaling factor to vary from stick to slip contact. The sliding traction is
</p>

<blockquote>
<code>S<sub>slide</sub> = k &eta; &Delta;v</code>
</blockquote>

<p>where <i>k</i> is a scaling factor (with units 1/length) and &eta; is viscosity of a fluid (which may depend on shear rate). Note that k = 0 leads to zero sliding traction or pure slipe; k large approaches stick contact; all other values of k give partial-slip boundary conditions that depend on the liquid's viscosity and could be tailored to match experimental results. The properties for this law are:</p>

<ul>
<li><code>coeff</code> for the scaling factor <i>k</i> in the contact law (default is 2 <a href="commands.html#units">(length units)<sup>-1</sup></a>).</li>
<li><code>LiquidPhase</code> for the liquid phase material by ID (default is none).</li>
<li><code>displacementOnly</code> 0 for contact by separation and stress, 1 for contact by separation only, or &lt;0 for contact by sepaction and interfacial stress &lt; <code>-displacementOnly</code> (in <a href="commands.html#units">pressure units/length units</a>) (default is 0).</li>
<li><code>Dc</code> &lt;0 to find separation assuming perfect interfaciall contact, &ge;0 to use <a href="#linearimp">imperfect interface methods</a> to find separation (in <a href="commands.html#units">pressure units/length units</a>) (default is -1)</li>
</ul>

<p>Note that this contact law uses the viscosity provided by the entered <code>LiquidPhase</code> material for all contact situations.</p>

<p>|<a href="#tables">Material Tables</a>|</p>

<!-- Material  Section -->

<a name="linearimp"></a><h2>Linear Imperfect Interface Contact Law (Type=62)</h2>

<p>This law implements an imperfect interface with tractions that depend linearly on displacement discontinuities at the interface, although slopes can be different in tension or compression:
</p>

<blockquote>
<code>T<sub>N</sub> = D<sub>nt</sub>&delta;<sub>N</sub></code> &nbsp;&nbsp;&nbsp;&nbsp;when separated<br />
<code>T<sub>N</sub> = D<sub>nc</sub>&delta;<sub>N</sub></code> &nbsp;&nbsp;&nbsp;&nbsp;when in contact<br />
<code>T<sub>S</sub> = D<sub>t</sub>&delta;<sub>S</sub></code>
</blockquote>

<p>The properties are the three interface parameters:</p>

<ul>
<li>Dnt - the interfacial stiffness in tension (in <a href="commands.html#units">pressure units/length units</a>) (default -1)</li>
<li>Dnc - the interfacial stiffness in compression (in <a href="commands.html#units">pressure units/length units</a>). This property is optional and lets you specify a different stiffness for compression (interfaces in contact) then for tension (interfaces separated). If <code>Dnc</code> is omitted, the stiffnesses will be the same in tension and compression (<i>i.e.</i>,  <code>Dnc</code> equal to <code>Dnt</code>).</li>
<li>Dt - the interfacial stiffness in shear (in <a href="commands.html#units">pressure units/length units</a>) (default -1)</li>
</ul>

<p>Although an interface becomes perfect as the interface parameters approach infinity, high interfacial stiffness make the problem numerically unstable. To model a perfect interface in any direction, set the interface parameter to -1, instead of using a high value.
</p>

<p>|<a href="#tables">Material Tables</a>|</p>

<!-- Material  Section -->

<a name="nonlinearimp"></a><h2>Nonlinear Imperfect Interface Contact Law (Type=65)</h2>

<p>This law implements an imperfect interface with tractions that depend on displacement discontinuities at the interface. It is currently the same as the <a href="#linearimp">linear interface law</a> except the implementation is done differently. It is more approximate, but likely very close to linear law and possibly faster. The parameters are:</p>

<ul>
<li>Dnt, Dnc, and Dt - see <a href="#linearimp">linear interface law</a></li>
<li>order - sets level of approximation to 0<sup>th</sup> order or 1<sup>st</sup> order (default 1)</li>
</ul>

<p>In the future, this law (or subclasses of this law) will add actual non-linear traction laws.</p>

<p>|<a href="#tables">Material Tables</a>|</p>

<!-- Material  Section -->

<h2><a name="linear"></a>Linear Hardening (number=1)</h2>

<p>In linear hardening the yield stress is given by
</p>

<blockquote>&sigma;<sub>Y</sub> = &sigma;<sub>Y0</sub> + E<sub>p</sub>&alpha; 
   = &sigma;<sub>Y0</sub>(1 + K&alpha;)
</blockquote>

<p>where <code>&sigma;<sub>Y0</sub></code> is the initial yield stress, <code>Ep</code> is the plastic modulus, <code>K</code> is hardening coefficient, and <code>&alpha;</code> is cumulative, equivalent plastic strain. The input properties are:</p>

<ul>

<li>yield - the initial yield stress (in <a href="commands.html#units">pressure units</a>).</li>
<li>Ep - the plastic modulus (in <a href="commands.html#units">pressure units</a>). The default is 0.0 which results in an elastic-perfectly plastic material. Ep can only be non-negative. To modeling softening with negative Ep, omit this parameter and enter a negtative Khard instead.</li>
<li>Khard - alternatively, you can enter this dimensionless parameter for hardening. It is only used if Ep is not entered.</li>
<li>yieldMin - the minimum yield stress (in <a href="commands.html#units">pressure units</a> with default of 0). A minimum stress is only used when modeling softening by entering <code>Khard</code> &lt; 0.</li>

</ul>

<h3>History Data for Linear Hardening</h3>

<ol>
<li>The cumulative equivalent plastic strain, <code>&alpha;</code>, defined as the sum of <code>sqrt(2/3)||d&epsilon;<sup>p</sup>||</code>.</li>
</ol>

<p>|<a href="#tables">Material Tables</a>|</p>

<!-- Material  Section -->
 
<h2><a name="nonlinear"></a>Nonlinear Hardening (number=2)</h2>

<p>In nonlinear hardening the yield stress is given by</p>

<blockquote>&sigma;<sub>Y</sub> = &sigma;<sub>Y0</sub>(1 + K&alpha;)<sup>n</sup>
</blockquote>

<p>where <code>K</code> and <code>n</code> are dimensionless hardening parameters. The input properties are:</p>

<ul>
<li>yield - the initial yield stress (in <a href="commands.html#units">pressure units</a>).</li>
<li>Khard - coefficient for nonlinear hardening (dimensionless).</li>
<li>nhard - exponent parameter in the nonlinear hardening (dimensionless).</li>
<li>yieldMin - the minimum yield stress (in <a href="commands.html#units">pressure units</a> with default of 0). A minimum stress is only used when modeling softening by entering <code>Khard</code> &lt; 0.</li>
</ul>

<h3>History Data for Nonlinear Hardening</h3>

<ol>
<li>The cumulative equivalent plastic strain, <code>&alpha;</code>, defined as the sum of <code>sqrt(2/3)||d&epsilon;<sup>p</sup>||</code>.</li>
</ol>

<p>|<a href="#tables">Material Tables</a>|</p>

<!-- Material  Section -->

<h2><a name="nonlinear2"></a>Alternate Nonlinear Hardening (number=6)</h2>

<p>In this alternate nonlinear hardening the yield stress is given by</p>

<blockquote>&sigma;<sub>Y</sub> = &sigma;<sub>Y0</sub>(1 + K&alpha;<sup>n</sup>)
</blockquote>

<p>where <code>K</code> and <code>n</code> are dimensionless hardening parameters. The input properties and history data are the same as for the <a href="#nonlinear">other nonlinear hardening law</a>.</p>

<p>|<a href="#tables">Material Tables</a>|</p>

<!-- Material  Section -->

<h2><a name="johnsoncook"></a>Johnson-Cook Hardening (number=3)</h2>

<p>The Johnson-Cook hardening law is:
</p>
<blockquote>
     σ<sub>y</sub> = (Ajc + Bjc ε<sub>p</sub><sup>njc</sup>) [1 + Cjc ln(dε<sub>p</sub>/ep0jc) + Djc ln(max(dε<sub>p</sub>/ep0jc,1))<sup>n2jc</sup>] (1 - Tr<sup>mjc</sup>)
</blockquote>
<p>where &epsilon;<sub>p</sub> is equivalent plastic strain, d&epsilon;<sub>p</sub> is plastic strain rate, and the reduced temperature (Tr) is given by</p>
<blockquote>
     Tr = (T - T<sub>0</sub>) / (Tmjc - T<sub>0</sub>)
</blockquote>
<p> and T<sub>0</sub> is the current <a href="commands.html#thermal">stress free temperature</a>. The law properties are:</p>

<ul>
<li>Ajc - the initial yield stress at reference rate and temperature (in <a href="commands.html#units">pressure units</a>).</li>
<li>Bjc - hardening term in the yield stress (in <a href="commands.html#units">pressure units</a>).</li>
<li>njc - exponent for power-law hardening term (dimensionless).</li>
<li>Cjc - coefficient for rate-dependent term (dimensionless).</li>
<li><Djc - coefficient for second rate-dependent term. (dimensionless)</li>
<li>n2jc - exponent on second rate-dependent term with D (dimensionless).</li>
<li>ep0jc - reference strain rate for reference yield stress in <code>Ajc</code> (in <a href="commands.html#units">1/time units</a>). (default is 1)</li>
<li>Tmjc - the material's melting point (in K).</li>
<li>mjc - exponent in the thermal term (dimensionless).</li>
</ul>

<h3>History Data for Johnson-Cook Material</h3>

<ol>
<li>The cumulative equivalent plastic strain, <code>&alpha;</code>, defined as the sum of <code>sqrt(2/3)||d&epsilon;<sup>p</sup>||</code>.</li>
</ol>

<p>|<a href="#tables">Material Tables</a>|</p>

<!-- Material  Section -->

<h2><a name="scgl"></a>Steinberg-Cochran-Lund Hardening (number=4)</h2>

<p>In this hardening law, both the shear modulus and the yield stress change. The shear modulus  is given by
</p>
<blockquote>
     G = G<sub>0</sub> [ 1 + (G<sub>p</sub>'/G<sub>0</sub>)(P/&eta;<sup>1/3</sup>) + (G<sub>T</sub>'/G<sub>0</sub>)(T-T<sub>0</sub>) ]
</blockquote>
<p> where &eta; = 1/(1-x) and x = -&Delta;V/V<sub>0</sub>. Here T<sub>0</sub> is the reference temperature for the material that is set using the <a href="commands.html#thermal"><code>StressFreeTemp</code></a> command.
This yield stress is given by:
</p>
<blockquote>
     &sigma;<sub>y</sub> = &sigma;<sub>0</sub> (1 + &beta; &epsilon;<sub>p</sub><sup>n</sup>) G
</blockquote>
<p>where &epsilon;<sub>p</sub> is the equivalent plastic strain and G is current shear modulus from above. The yield stress is limited to a maximum yield stress.
The properties for this law are:</p>

<ul>
<li>yield - the initial yield stress (&sigma;<sub>0</sub>) at zero pressure and the reference temperature (in <a href="commands.html#units">pressure units</a>).</li>
<li>yieldMax - maximum yield stress (in <a href="commands.html#units">pressure units</a>).</li>
<li>betahard - yield stress hardening term (dimensionless).</li>
<li>nhard - exponent on cumulative plastic strain in hardening term (dimensionless).</li>
<li>GPpG0 - G<sub>p</sub>'/G<sub>0</sub> term for pressure dependence of shear modulus (in units <a href="commands.html#units">1/pressure units</a> or 0 to omit pressure dependence in shear modulus).</li>
<li>GTpG0 - G<sub>T</sub>'/G<sub>0</sub> term for temperature dependence of shear modulus (in units K<sup>-1</sup>) or 0 to omit temperature dependence in shear modulus.</li>
</ul>

<h3>History Data for SCGL Hardening Law</h3>

<ol>
<li>The cumulative equivalent plastic strain, <code>&alpha;</code>, defined as the sum of <code>sqrt(2/3)||d&epsilon;<sup>p</sup>||</code>.</li>
</ol>

<p>|<a href="#tables">Material Tables</a>|</p>

<!-- Material  Section -->

<h2><a name="SL"></a>Steinberg-Lund Hardening Law (number=5)</h2>

<p>This hardening law has a rate- and temperature-dependent yield stress given by:
</p>
<blockquote>
     &sigma;<sub>y</sub> = {Y<sub>T</sub>(d&epsilon;<sub>p</sub>/dt,T) + &sigma;<sub>0</sub>(1 + &beta; &epsilon;<sub>p</sub><sup>n</sup>)} G
</blockquote>
<p>This law is identical to the <a href="#SCGL">SCGL law</a> except for the new rate- and temperature-dependent term Y<sub>T</sub>(d&epsilon;<sub>p</sub>/dt,T), which is only defined in inverse form:</p>
<blockquote>
d&epsilon;<sub>p</sub>(Y<sub>T</sub>,T)/dt = { (C<sub>2</sub>/Y<sub>T</sub>) + (1/C<sub>1</sub>)exp[(2U<sub>k</sub>/(kT))(1-(Y<sub>T</sub>/Y<sub>P</sub>))<sup>2</sup>] }<sup>-1</sup>
</blockquote>
<p>where &epsilon;<sub>p</sub> is the equivalent plastic strain and Y<sub>T</sub> is limited to &le; Y<sub>P</sub>. The properties for this law are:
</p>

<ul>
<li>C1SL - hardening law constant (C<sub>1</sub>) (in <a href="commands.html#units">1/time units</a>).</li>
<li>C2SL - hardening law constant (C<sub>2</sub>) (in <a href="commands.html#units">pressure units-time units</a>c).</li>
<li>YP - the Peierls stress and also the maximum rate-dependent yield stress (in <a href="commands.html#units">pressure units</a>).</li>
<li>UkOverk - an energy associated with forming kinks (enter Uk/k in K).</li>
<li><a href="#SCGL">(other)</a> - enter all other properties required for a <a href="#SCGL">SCGL Hardening Law</a></li>
</ul>

<h3>History Data for Steinberg-Lund Hardening Law</h3>

<ol>
<li>The cumulative equivalent plastic strain, <code>&alpha;</code>, defined as the sum of <code>sqrt(2/3)||d&epsilon;<sup>p</sup>||</code>.</li>
<li>The current rate- and temperature-dependent yield stress (Y<sub>T</sub>) in <a href="commands.html#units">pressure units</a>.</li>
<li>The current equivalent plastic strain rate (d&epsilon;<sub>p</sub>/dt in 1/sec).</li>
</ol>

<p>|<a href="#tables">Material Tables</a>|</p>

<!-- Material  Section -->

<h2><a name="maxPS"></a>Maximum Principle Stress (number=1)</h2>

<p>The damage initiates when either the maximum principle stress exceeds the tensile strength or the maximum shear stress exceeds the shear strength. The input properties are:</p>

<ul>

<li>sigmac - the tensile strength (in <a href="commands.html#units">pressure units</a>). The default is very large number which means the material will not fail in tension.</li>
<li>tauc - the shear strength (in <a href="commands.html#units">pressure units</a>). The default is very large number which means the material will not fail in shear.</li>
</ul>

<p>|<a href="#tables">Material Tables</a>|</p>

<!-- Material  Section -->

<h2><a name="tiFailure"></a>Transversely Isotropic Failure Initiation (number=2)</h2>

<p>The damage initiates when axial stress exceeds axial tensile strength, axial shear stress exceeds axial shear strength, principle stress in the transverse plane exceeds transverse shear strength, or maximum shear stress in transverse plane exceeds the shear strength.</p>

<ul>
<li>sigmacA - critical axial strength (inMPa) for failure in the axial direction</li>
<li>sigmac - critical transverse tensile strength (in <a href="commands.html#units">pressure units</a>) for tensile failure in the isotropic plane</li>
<li>taucA - critical shear strength (in <a href="commands.html#units">pressure units</a>) for failure due to axial shear stress with failure parallel to the axial direction</li>
<li>taucT - Critical shear strength (in <a href="commands.html#units">pressure units</a>) for failure due to axial shear stress with failure through the axial direction</li>
<li>tauc  - critical transverse shear strength (in <a href="commands.html#units">pressure units</a>) for shear failure in the isotropic plane</li>
</ul>

<p>All strength values default to a large number, which means failure will not occur in that mode.</p>

<p>|<a href="#tables">Material Tables</a>|</p>

<!-- Material  Section -->

<h2><a name="linearSoft"></a>Linear Softening (number=1)</h2>

<p>The linear softening law is f(&delta;) = 1 - &delta;/&delta;<sub>max</sub>. The input property is:</p>

<ul>

<li>Gc - the toughness of the material is this law's failure mode (in <a href="commands.html#units">energy release units</a>). The default is very large number which means the material will not soften after initiation of damage.</li>
</ul>

<p>Note that you do not enter &delta;<sub>max</sub>. It is determined at run time because it is affected by particle volume, initiation stress, and crack orientation.</p>

<p>|<a href="#tables">Material Tables</a>|</p>

<!-- Material  Section -->

<h2><a name="expSoft"></a>Exponential Softening (number=2)</h2>

<p>The exponential softening law is f(&delta;) = exp(-k&delta;). The input property is:</p>

<ul>

<li>Gc - the toughness of the material is this law's failure mode (in <a href="commands.html#units">energy release units</a>). The default is very large number which means the material will not soften after initiation of damage.</li>
</ul>

<p>Note that you do not enter <i>k</i>. It is determined at run time because it is affected by particle volume, initiation stress, and crack orientation.</p>

<p>|<a href="#tables">Material Tables</a>|</p>

<!-- Material  Section -->

<h2><a name="cubicStep"></a>Cubic Step Function Softening (number=3)</h2>

<p>The cubic step function softening law is f(&delta;) = 1 + (&delta;/&delta;<sub>max</sub>)<sup>2</sup>(2(&delta;/&delta;<sub>max</sub>)-3.). The input property is:</p>

<ul>

<li>Gc - the toughness of the material is this law's failure mode (in <a href="commands.html#units">energy release units</a>). The default is very large number which means the material will not soften after initiation of damage.</li>
</ul>

<p>Note that you do not enter &delta;<sub>max</sub>. It is determined at run time because it is affected by particle volume, initiation stress, and crack orientation.</p>

<p>|<a href="#tables">Material Tables</a>|</p>
<!-- Material  Section -->

<h2><a name="common"></a>Material Properties Common For All Material Types</h2>

<p>There are some material properties that are common to all material types, although not all materials will use them all. The categories of common properties are:
</p>

<ul>
<li><a href="#cmn1">Basic Properties</a></li>
<li><a href="#cmn2">Fracture Toughness Properties</a></li>
<li><a href="#cmn3">Crack Propagation Properties</a></li>
<li><a href="#cmn4">Contact Properties</a></li>
<li><a href="#cmn5">Artificial Viscosity</a></li>
<li><a href="#cmn6">Poroelasticity Properties</a></li>
</ul>


<h3><a name="cmn1"></a>Basic Properties</h3>

<p>These are basic material properties</p>

<ul>
<li>rho - the density (in <a href="commands.html#units">density units</a>)</li>
<li>Cv - the constant strain heat capacity (in <a href="commands.html#units">heat capacity units</a>). You do not need to enter a constant pressure heat capacity (C<sub>p</sub>) because it is not needed. Some materials will output C<sub>p</sub> by using thermodynamic relations for the difference between C<sub>p</sub> and C<sub>v</sub>, but C<sub>p</sub> is never needed in calculations.</li>
<li>kCond - the thermal conductivity for isotropic materials (in <a href="commands.html#units">conductivity units</a>) </li>
<li>csat - the saturation concentration entered as a weight fraction</li>
<li>beta - the concentration expansion coefficient for isotropic materials (in strain/(wt fraction))</li>
<li>D - the diffusion constant for isotropic materials (in <a href="commands.html#units">diffusion units</a>)</li>
<li>largeRotation - to use (1) or not use (0, the default) polar decomposition when evaluating rotations. This option is only allowed for linear elastic and elastic-plastic small strain materials. Option 1 uses polar decomposition to evaluate incremental rotation and uses that rotation to update strains and stresses. Option 0 use approximations of polar decomposition to rotate strain and stress updates (standard hypoelasticity). Both methods work for large rotations (provided the rotations are incrementally small). Option 1 may be more accurate while options 0 is more efficient.</li>
<li>matDamping - sets custom particle damping to apply only to particles of this material. Its value replaces global <a href="commands.html#mhother">particle damping</a> setting and must be a constant (function of time not allowed).</li>
<li>matPIC - sets custom PIC damping to apply only to particles of this material. Its value replaces the global <a href="commands.html#mhdamp">PIC damping</a> setting. Note that to use custom <a href="commands.html#mhdamp">XPIC damping</a> in materials, the global settings must activate XPIC and then the <code>matPIC</code> property can change a material to use a different amount of XPIC or switch to FLIP. If the global settings are 100% FLIP, the <code>matPIC</code> property can implement PIC, but not XPIC.</li>
<li>color (red),(green),(blue),(alpha) - sets the color of the material with the four arguments being RGB and alpha values between 0.0 and 1.0. A single argument can be used instead to set gray level between 0.0 and 1.0 (with alpha=1.0). Three arguments set red, green, and blue with alpha=1.0. If no color is provided, a color will be picked from the current spectrum using the material number.</li>
</ul>

<h3><a name="cmn2"></a>Fracture Toughness Properties</h3>

<p>These properties set material properties that determine the fracture toughness of the material and control various aspects of crack propagation. Note that whether or not the properties are used depends on the <a href="commands.html#cracks">selected criterion for crack growth</a>.</p>

<ul>
<li>JIc or JIc - critical mode I or mode II energy release rate (in <a href="commands.html#units">energy release units</a>).</li>
<li>KIc or KIIc - critical mode I or mode II stress intensity factor (in <a href="commands.html#units">stress intensity units</a>).</li>
<li>KIexp or KIIexp - the exponent p or q in the elliptical criteria for crack growth.</li>
<li>delIc or delIIc - critical crack opening displacements in mode I and mode II.</li>
<li>initTime - time to start crack growth (in <a href="commands.html#units">alt time units</a>).</li>
<li>speed - define the constant crack speed (in <a href="commands.html#units">velocity units</a>).</li>
<li>maxLength - define the maximum length of the crack (in <a href="commands.html#units">length units</a>). The crack growth will stop when this length is reached.</li>
<li>nmix - a parameter used in some mixed-mode failure laws.</li>
<li>gamma, p, or gain - still in development and not meant for general use.</li>
</ul>

<h3><a name="cmn3"></a>Crack Propagation Properties</h3>

<p>The criterion, direction, and traction properties (and the analogous alternate propagation properties) are set with following XML commands (instead of standard property commands):</p>

<ul>
<li>criterion - set crack growth criterion for this material
(see options in <a href="commands.html#csettings">Propagate Command</a>).</li>

<li>direction - set optional crack growth direction for this material
(see options in <a href="commands.html#csettings">Propagate Command</a>).</li>

<li>traction - set optional traction law to create for crack propagation in this material by ID or number (see <a href="commands.html#csettings">Propagate Command</a> for details).</li>

<li>altcriterion is identical to criterion, but applies instead to the alternate crack growth criterion in the <a href="commands.html#csettings">AltPropagate Command</a>.</li>

<li><code>altdirection</code> is identical to <code>direction</code>, but applies instead to the alternate crack growth direction in the <a href="commands.html#csettings">AltPropagate Command</a>.</li>

<li><code>alttraction</code> is identical to <code>traction</code>, but applies instead to the alternate crack growth traction law material in the <a href="commands.html#csettings">AltPropagate Command</a>.</li>

<li>xGrow and yGrow -  specify a unit vector for a constant crack growth direction. They only apply for crack propagation with <a href="commands.html#csettings">criterion 2</a> and then only when that criterion selects the default propagation direction.</li>

<li>constantTip - set to 0 or 1 where 1 means the crack tip will always uses its initial material even if the crack is in or moves to another material</li>
</ul>


<h3><a name="cmn4"></a>Contact Properties</h3>

<p>These commands allow custom contact properties for each material pair
in <a href="commands.html#mmmode">multimaterial mode MPM</a>.</p>

<ul>

<li><code>Contact</code> - a Contact material property can define custom contact law
for <a href="commands.html#mmmode">multimaterial mode MPM</a> contact between the current material
and another material.</li>

<li><code>shareMatField (matID)</code> - set to another material (by material ID) to share material velocity field
with that material. Two (or more) materials sharing the same velocity field will move with perfect
contact (<i>i.e.</i>, in a single velocity field) among themselves, but can interact with other
materials in different velocity fields by contact or interface laws. Materials sharing velocity fields must be compatible (<i>e.g.</i>, all rigid or all nonrigid) and the material in <code>(matID)</code> must not be sharing it's field.</li>

</ul>

<p>The <a href="commands.html#matprops">deprecated material properties</a> <code>Friction</code> and <code>Interface</code> should be replaced by the above <code>Contact</code> property.</p>

<h3><a name="cmn5"></a>Artificial Viscosity</h3>

<p>Artificial viscosity adds a pressure, Q, related to velocity gradient on the particle, but only when it is compressing. The equation is:</p>
<blockquote>
Q = &Delta;x*|D<sub>kk</sub>|*(A1*C + A2*&Delta;x*|D<sub>kk</sub>|)
</blockquote>
<p>where &Delta;x is the cell size of the mesh, |D<sub>kk</sub>| is the relative volume change rate, C is the bulk wave speed in the material, and A1 and A2 are adjustable constants.</p>

<ul>
<li>ArtificialVisc - set to "on" or "off" to determine if artificial viscosity is on.</li>
<li>avA1 - the A1 parameter in artificial viscosity (dimensionless, default 0.2)</li>
<li>avA2 - the A2 parameter in artificial viscosity (dimensionless, default 2.0)</li>
</ul>
<p>The artificial viscosity option is only supported in some materials (as documented <a href="http://osupdocs.forestry.oregonstate.edu/index.php/Common_Material_Properties#Artificial_Viscosity">here</a>). If you use these commands in a material that does not support, an error will result.</p>

<h3><a name="cmn6"></a>Poroelasticity Properties</h3>

<p>Some materials support poroelasticity calculations and the properties in this section control pore pressure flow between particles and coupling between stress and strain and pore pressure. The properties to use depend on symmetry of the parent material.</p>

<p><b>Isotropic Poroelasticity Properties</b></p>

<ul>
<li>Ku - undrained bulk modulus (<a href="commands.html#units">pressure units</a>)</li>
<li>alphaPE - The poroelasticity Biot coefficient (from 0 to 1) (dimensionless, default 0)</li>
<li>Darcy - Darcy law permittivity for the material (1/<a href="commands.html#units">length units<sup>2</sup></a>, default 0)</li>
</ul>

<p><b>Transversely Isotropic Poroelasticity Properties</b></p>

<ul>
<li>Ku - undrained bulk modulus (<a href="commands.html#units">pressure units</a>)</li>
<li>alphaAPE, alphaTPE - The axial and transverse poroelasticity Biot coefficients (from 0 to 1) (dimensionless, default 0)</li>
<li>DarcyA, DarcyT - Darcy axial and transverse law permittivities for the material (1/<a href="commands.html#units">length units<sup>2</sup></a>, default 0)</li>
</ul>

<p><b>Orthotropic Poroelasticity Properties</b></p>

<ul>
<li>Ku - undrained bulk modulus (<a href="commands.html#units">pressure units</a>)</li>
<li>alphaxPE, alphayPE, alphazPE - The poroelasticity Biot coefficients in the three orthrotopic directions (from 0 to 1) (dimensionless, default 0)</li>
<li>alphaRPE, alphaZPE, alphaTPE - For cylindrical, orthotropic materials, these are radial, axial, and hoop Biot coefficients. They are synonyms for orthotropic coefficients (with R=x, Z=y, and T=z). (from 0 to 1) (dimensionless, default 0)</li>
<li>Darcyx, Darcyy, Darcyz - Darcy law permittivities for the three orthotropic directions (1/<a href="commands.html#units">length units<sup>2</sup></a>, default 0)</li>
<li>DarcyR, DarcyZ, DarcyT - For cylindrical, orthotropic materials, these are radial, axial, and hoop Darcy law permittivities for the material. They are synonyms for orthotropic coefficients (with R=x, Z=y, and T=z). (1/<a href="commands.html#units">length units<sup>2</sup></a>, default 0)</li>
</ul>

<p>|<a href="#tables">Material Tables</a>|</p>

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